Boundses

Boundses are a mathematical construct intended to describe envelope-like bounds for families of real-valued functions defined on a common domain. They generalize the classical notion of a bound by providing a single nonnegative function that dominates every member of a specified function class pointwise.

Given a domain D and a class F of functions f: D → R, a bounds is a function b: D → [0, ∞) such that for every f in F and all x in D, |f(x)| ≤ b(x). The collection of all such b is the bounds of F; if there is a pointwise smallest one, it is called the envelope of F.

Construction typically uses the pointwise supremum: b(x) = sup_{f ∈ F} |f(x)| when this is finite. The envelope is measurable if F consists of measurable functions and is unique up to almost everywhere equality. If F is uniformly bounded by M, then b ≡ M is a simple envelope.

Examples include a family F of all functions on a compact interval with sup norms bounded by 1, yielding the constant envelope b(x) ≡ 1. For function families with additional growth or smoothness constraints, envelopes can often be obtained from known majorants.

Applications of boundses appear in approximation theory, numerical analysis, and learning theory, where they provide uniform error bounds, stability estimates, and generalization guarantees.

The term boundses is a plural form of bounds used to discuss envelopes of function classes, connecting to classical ideas of majorants and envelopes in analysis.